The Banach spaces ℓ∞(c0) and c0(

Abstract

The statement of the title is proved. It follows from this that the spaces c 0 ( p ), p (c 0 ) and p ( q ), 1 p, q +∞, make a family of mutually non-isomorphic Banach spaces.In this note we use standard notation in Banach space theory as in [1] or [5].Some months ago Félix Cabello asked us: are the Banach spaces ∞ (c 0 ) and c 0 ( ∞ ) isomorphic? In this note we show that the answer is "no". In fact, we will prove that ∞ (c 0 ) is not even isomorphic to a quotient of c 0 ( ∞ ). We will also get some consequences of this result. To prove it we need a preliminary lemma.Lemma 1. Let W be a relatively weakly compact subset of c 0 and 0 < λ < 1, then there exists a norm one vector y ∈ c 0 such that